Question

Consider the following statements
Statement I The solution set of the inequalities $x+2 y \leq 8,2 x+y \leq 8, x \geq 0, y \geq 0$ is

Statement II The solution set of the inequalities $x \geq 3$ and $y \geq 2$ is

Choose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both are true
• D. Both are false

Answer 1

Answer: (A)

I. We draw the graphs of the lines $x+2 y=8$ and $2 x+y=8$. The inequality (1) and (2) represent the region below the two lines, including the point on the respective lines.

Since, $x \geq 0, y \geq 0$, every point in the shaded region in the first quadrant represent a solution of the given system of inequalities.
II. The given system of inequalities
$ x \geq 3 $ ...(i)
$ y \geq 2$ ....(ii)
Step I Consider the inequations as strict equations i.e., $x=3, y=2$
Step II Plot the graph. The graph of $x=3$, (i.e., $y=0$ ) will be a line parallel to $Y$-axis, intersecting the $X$-axis at 3 . The graph of $y=2$ (i.e., $x=0$ ) will be a line parallel to $X$-axis, intersecting the $Y$-axis at 2 .Step III Take a point $(0,0)$ and put it in the given inequations (i) and (ii).
$\text { i.e., } 0 \geq 3 \text { (false) }$
$0 \geq 2 \text { (false) }$

$\therefore$ Both the graphs will be parallel and away from the origin.
Thus, common shaded region shows the solution of the inequalities.